Field · 3 models
Network Simulations
When who is connected to whom matters, averages over a whole population stop being enough. These models build networks, spread infection over them and rank their nodes, comparing each simulation with network theory.
Models in this field
- OS-031Random network (Erdős–Rényi)Link every pair at random: Poisson degrees, and a giant component that appears suddenly at mean degree 1.
- OS-032Epidemic on a networkSIR spreading along the edges of random, small-world and scale-free networks, compared with homogeneous mixing.
- OS-033PageRankRank pages by where a random surfer spends its time: power iteration vs a simulated walk.
The models
- Random network: the Erdős–Rényi graph and its sudden phase transition — a giant connected component appears when the mean degree passes 1.
- Epidemic on a network: the SIR model with contacts along edges, where clustering and hubs change how far and how fast infection spreads.
- PageRank: node importance as the long-run share of time a random surfer spends on each page, computed two ways.
Structure changes dynamics
A homogeneous-mixing model such as the SIR equations assumes every pair of people is equally likely to meet. On a network, an infected person can only infect neighbors, neighbors of neighbors are often already infected, and highly connected nodes are infected early and spread widely. The same transmission and recovery rates can produce very different outbreaks on different networks.
What to measure on a network
A handful of quantities describe most networks: the degree distribution (how many neighbors each node has, and how unequal that is), the clustering coefficient (how often two neighbors of a node are themselves connected), the size of the largest connected component, and typical path lengths. Random graphs have low clustering and short paths; real social networks usually have high clustering and short paths at the same time — the small-world property that the Watts–Strogatz option in the epidemic model reproduces. Each page reports the measures its model depends on alongside the theoretical value for comparison.
PageRank ties networks back to Markov chains: the random surfer is a Markov chain on the pages, and the ranking is its stationary distribution.